Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Wallis's ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10.As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.
The motion of bubbles dispersed in a liquid when a small-amplitude oscillatory motion is imposed on the mixture is examined in the limit of small frequency and viscosity. Under these conditions, for bubbles with a stress-free surface, the motion can be described in terms of added mass and viscous force coefficients. For bubbles contaminated with surface-active impurities, the introduction of a further coeflicient to parametrize the Basset force is necessary. These coefficients are calculated numerically for random configurations of bubbles by solving the appropriate multibubble interaction problem exactly using a method of multipole expansion. Results obtained by averaging over several configurations are presented. Comparison of the results with those for periodic arrays of bubbles shows that these coefficients are, in general, relatively insensitive to the detailed spatial arrangement of the bubbles. On the basis of this observation, it is possible to estimate them via simple formulas derived analytically for dilute periodic arrays. The effect of surface tension and density of bubbles (or rigid particles in the case where the no-slip boundary condition is applicable 1 is also examined and found to be rather small.
Hiperco® 50 alloy heat treated between 450 and 650 °C exhibits superior mechanical properties. We report the measurements of the ac core loss at various frequencies up to 4500 Hz of the Hiperco® 50 alloy samples annealed at 450 and 650 °C. The 650 °C annealed specimens have lower ac core loss than that of the 450 °C annealed ones. The total core loss, consisting of contributions from hysteresis core loss and eddy-current core loss, depends on frequency f as af+bf2. The eddy-current loss of a single laminate is minor compared to the hysteresis loss.
The motion of spheres (bubbles) in an incompressible liquid undergoing a small-amplitude oscillatory motion is calculated by a multipole expansion method. In the limit of small viscosity, the Stokes layer is confined to the vicinity of the surface of the bubbles, which therefore interact approximately only through the pressure field. The motion of the spheres is parametrized in terms of added mass, Basset, and drag forces; the coefficients of which are obtained from the simulation. To obtain results useful for the study of pressure wave propagation in bubbly liquids, several bubble configurations are studied for different (finite) volume fractions and the results then averaged. The effects of surface tension and bubble density are also considered. [Work supported by DOE.]
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